Right Operation is Entropic

Theorem

The right operation is entropic:

$\forall a, b, c, d: \paren {a \to b} \to \paren {c \to d} = \paren {a \to c} \to \paren {b \to d}$

Proof

\(\ds \paren {a \to b} \to \paren {c \to d}\)

\(=\)

\(\ds b \to d\)

Definition of Right Operation

\(\ds \)

\(=\)

\(\ds d\)

\(\ds \paren {a \to c} \to \paren {b \to d}\)

\(=\)

\(\ds c \to d\)

Definition of Right Operation

\(\ds \)

\(=\)

\(\ds d\)

\(\ds \)

\(=\)

\(\ds \paren {a \to b} \to \paren {c \to d}\)

a priori

$\blacksquare$

Also see

• Left Operation is Entropic

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Exercise $13.12 \ \text{(c)}$